Why do many Ashkenazic congregations repeat (at least a portion of) the last verse (Devarim 25:19) of Parashat Zachor twice, once saying "zeikher" and the other "zekher" (flipping the first vowel from tzeirei to segol)? Why do many such congregations have a similar practice for two words in the Scroll of Esther, in verse 8:11 ("v'laharog" / "laharog") and verse 9:2 ("lifneihem" / "bifneihem")? How did these doubts regarding the correct versions arise, and why do Sefardim and Teimanim not seem to have such doubts? What are the arguments for and against repeating these words? For those that do repeat the words, does it matter which version is said first? To what extent do these issues affect the validity of the performance of the mitzvot to hear these readings? (For that matter, is it even humanly possible to correctly perform the mitzva of hearing the megilla?)
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There are actually many such uncertainties throughout Tanach - some on vowels, others on the spelling of words.
As far as I know, the reason we're more particular about accommodating all opinions in Parshas Zachor is because according to many opinions, hearing it is a Biblical obligation (Shulchan Aruch, Orach Chaim 685:7).
As for the Megillah reading - while Shalom is correct that we're more tolerant of mistakes in it, that may not apply to actual differences of wording (see Shulchan Aruch, Orach Chaim 690:14, and Beur Halachah there s.v. ein medakdekin).
There is an article (in Hebrew) about the repetitions in the Megillah here. The author traces the custom back to the late 18th or early 19th century, when the Masoretic commentary Minchas Shai was published and found to conflict (in these two cases) with older Ashkenazic megillos; the result was this compromise.
This article in turn refers to another one, by Rabbi Mordechai Breuer, available here; he discusses this as well as the "zeicher/zecher" variants.
Both of these articles argue that the custom of repeating the word or phrase should be abandoned, since there is strong evidence of the correct mesorah in all of these cases. However, this position is not universally accepted.
If there's a doubt about what's the exact correct text, we try to do it both ways. If I recall correctly from Rabbi Welcher in Queens, we read the most-likely way first, followed by the alternative option. Everyone agrees that "zeicher", which the Sefardic and Yemenite Jews have in their text (and they know they're right, and aren't bothered by this silly Ashkenazic question), is the more-probable reading.
We try to do it both ways because we reasonably can, but it wouldn't be required. For Zachor, I'd assume it follows the rules of regular Torah reading, in which if it doesn't change the meaning, it's still kosher. Megillah-reading is incredibly fault-tolerant (far more than Torah reading), so it would definitely be okay whether you said "no one stood up to them" instead of "no one stood up against them", or "to abolish, kill, and obliterate" instead of "to abolish and to kill and to obliterate."